Abstract
I put forward precise and appealing notions of reference, self-reference, and well-foundedness for sentences of the language of first-order Peano arithmetic extended with a truth predicate. These notions are intended to play a central role in the study of the reference patterns that underlie expressions leading to semantic paradox and, thus, in the construction of philosophically well-motivated semantic theories of truth.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Beall, J.C. (2001). Is Y ablo’s P aradox non-circular? Analysis, 61, 176–187.
Boolos, G., Burgess, J., Jeffrey, R. (2007). Computability and logic, 1edn. Cambridge: Cambridge University Press.
Carnap, R. (1937). Logische Syntax der Sprache. London: Routledge.
Cook, R.T. (2006). There are Non-circular Paradoxes (but Yablo’s Isn’t One of Them!). The Monist, 89(1), 118–149.
Gödel, K. (1931). Über formal unentscheidebarre Sätze der Principia Mathematica und verwandler S ystem I. Monathshefte für Mathematik und Physik, 38, 173–198.
Goodman, N. (1961). About. Mind, 70, 1–24.
Halbach, V., & Visser, A. (2014). Self-reference in Arithmetic I. Review of Symbolic Logic, 7, 671–691.
Halbach, V., & Visser, A. (2014). Self-reference in Arithmetic II. Review of Symbolic Logic, 7, 692–712.
Hardy, J. (1995). Is Y ablo’s P aradox liar-like? Analysis, 55(3), 197–198.
Heck, R.K. (2007). Self-reference and the languages of Arithmetic. Philosophia Mathematica III (pp. 1–29). Originally published under the name “Richard G. Heck Jr”.
Henkin, L. (1952). A problem concerning provability. Journal of Symbolic Logic, 17, 160.
Herzberger, H. (1970). Paradoxes of grounding in semantics. Journal of Philosophical Logic, 67, 145–167.
Jeroslow, R.G. (1973). Redundancies in the Hilbert-Bernays derivability conditions for Gödel’s second incompleteness theorem. Journal of Symbolic Logic, 38, 359–367.
Ketland, J. (2004). Bueno and C olyvan on Y ablo’s P aradox. Analysis, 64, 165–172.
Ketland, J. (2005). Yablo’s P aradox and ω-inconsistency. Synthese, 145, 295–307.
Kreisel, G. (1953). On a problem of H enkin’s. Indagationes Mathematicae, 15, 405–406.
Kripke, S. (1975). Outline of a theory of truth. Journal of Philosphy, 72, 690–716.
Leitgeb, H. (2002). What is a self-referential sentence? Critical remarks on the alleged (non)-circularity of Y ablo’s P aradox. Logique et Analyse, 177–178, 3–14.
Milne, P. (2007). On Godel̈ sentences and what they say. Philosophia Mathematica, III(15), 193–226.
Montague, R. (1962). Theories incomparable with respect to relative interpretability. Journal of Symbolic Logic, 27, 195–211.
Picollo, L. (2018). Reference in Arithmetic. Review of Symbolic Logic, 11, 573–603.
Picollo, L. Reference and truth. Journal of Philosophical Logic (to appear).
Priest, G. (1997). Yablo’s Paradox. Analysis, 57, 236–242.
Putnam, H. (1958). Formalization of the C oncept of “A bout”. Philosophy of Science, 25, 125–130.
Putnam, H. (1980). Models and Reality. Journal of Symbolic Logic, 45, 464–482.
Smoryński, C. (1991). The development of self-reference: Lob’s̈ theorem. In Drucker, T. (Ed.) Perspectives on the history of mathematical logic (pp. 110–133). Boston: Birkhäuser.
Urbaniak, R. (2009). Leitgeb, “About,” Yablo. Logique et Analyse, 207, 239–254.
Visser, A. (1989). Semantics and the liar paradox. In Gabbay, D. M., & Günthner, F. (Eds.) Handbook of philosophical logic, (Vol. 4 pp. 617–706). Dordrecht: Reidel.
Yablo, S. (1985). Truth and reflexion. Journal of Philosphical Logic, 14, 297–349.
Yablo, S. (1993). Paradox without self-reference. Analysis, 53, 251–252.
Yablo, S. (2014). Aboutness. Princeton: Princeton University Press.
Acknowledgements
I am deeply indebted to Volker Halbach, with whom I had countless fruitful discussions on reference and self-reference over the last seven years. I would also like to particularly thank Dan Waxman for extremely helpful comments on the final drafts, Thomas Schindler, for great suggestions and encouragement, and two anonymous referees for serious improvements in clarity and exposition. I should mention as well Eduardo Barrio, Catrin Campbell-Moore, Luca Castaldo, Roy T. Cook, Benedict Eastaugh, Martin Fischer, Hannes Leitgeb, Øystein Linnebo, Carlo Nicolai, Graham Priest, Johannes Stern, Albert Visser, the Buenos Aires Logic Group, the MCMP logic community, and the Oxford logic group. Finally, I would like to thank the Alexander von Humboldt Foundation and, especially, the Deutsche Forschungsgemeinschaft (DFG) for generously funding the research projects “Reference patterns of paradox” (PI 1294/1-1) and “The Logics of Truth: Operational and Substructural Approaches” (GZ HJ 5/1-1, AOBJ 617612).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Picollo, L. Alethic Reference. J Philos Logic 49, 417–438 (2020). https://doi.org/10.1007/s10992-019-09524-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-019-09524-w